Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493451 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1782 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62855 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3428 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,12,27)(4,15,18,7)(5,9)(6,10)(8,14,22,28)(13,29)(16,26)(17,24,32,21)(19,25)(20,31)(23,30);(2,7)(5,19)(8,20)(9,24)(10,15)(11,26)(14,28)(17,32)(21,25)(22,23)(27,29)(30,31);(1,3)(2,8,10,22)(4,16)(5,14)(6,12)(7,20,15,23)(9,11,25,27)(13,18)(17,31)(19,28)(21,29,24,26)(30,32);(3,12)(8,22)(11,27)(13,16)(14,28)(20,23)(26,29)(30,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fi,g^b=gh,d^c=di,e^c=ei,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400647);